Computational Multiphase Flow, TU Darmstadt
19 Oct, 2025
| Progress | Failure | |
|---|---|---|
| WP1 | Implementation of classical Poisson-Nernst-Planck electrokinetic transport - Segregated solver - Numerical stability Parametric Case Setup: - Pore geometry coupling with electrochemical dimensionless numbers |
Late times charging dynamics verification - Large computational cost |
| WP2 | Zero-flux boundary layer - Sub-grid zero-flux boundary layer |
Mass conservation testing - Benchmark incompatibility |
| WP3 | Extensible class inheritance structure - Modifications of Poisson-Nernst-Planck transport equations available |
|
| WP4 | ||
| WP5 |
Navier-Stokes Equations
\[ \frac{\partial}{\partial t}[\rho\mathbf{v}] + \nabla \cdot \{\rho\mathbf{v}\mathbf{v}\} = -\nabla p + \nabla \cdot \left\{\mu\left[\nabla\mathbf{v} + (\nabla\mathbf{v})^{\text{T}}\right]\right\} \]
\[ \frac{\partial\rho}{\partial t} +\nabla\cdot (\rho\mathbf{u}_{}) = 0 \]
Species Transport &
Gauss law
\[ \frac{ \partial c_{i} }{ \partial t } = \nabla \cdot(D_{i}\nabla c_{i}) + \nabla \cdot [( \underbrace{ D_{i} \dfrac{ez_{i}}{kT} } _{ \text{Stokes-Einstein relation} }\nabla \Psi)c_{i} ] \]
\[ \nabla^2 \Psi _{E} =\frac{-\rho_E}{\epsilon\epsilon_{0}} = \frac{-F \sum_{i} z_{i} c_{i}}{ \epsilon\epsilon_{0}} \]
Algorithm:
\[ \left[ \nabla \cdot c_{i} \right]_{f} \cdot n_{\perp} \mathbf{A}_{_{f}} = - \left[ \frac{\mu_{i}}{D_{i}}\left( \nabla \Psi \right)c_{i} \right]_{f} \cdot \mathbf{A}_{_{f}} \]
\[ \longrightarrow \nabla c_{i,P} + \frac{\mu_{i}}{D_{i}}\left( \nabla \Psi_{f} \right)c_{i,P} = 0 \]
$$
c_{i,P}^P = c_{i,f}^S ( ( { {f} }
-
{ {P} }
) )
$$
Not tested!
Implemented, but incompatible with benchmark described by 1
◦ Block matrix coupling scheme 3 (WP1)
Stability and cost
◦ Modified Poisson-Nernst-Planck variations (WP3)
Mass Conservation, transport dynamics near walls, moderate voltages